Coherence resonances in an excitable thermochemical system with multiple stationary states.

A master equation approach is used to study the influence of internal fluctuations on the dynamics of three excitable thermochemical systems exhibiting continuous as well as discrete changes of temperature. The systems differ by the types of excitability. The dependences of the relative deviations from mean values of the interspike intervals and escape times from the stable stationary state on the size of the systems calculated from simulations of stochastic trajectories exhibit minima, which testify to the appearance of resonance phenomena.

Oscillons localized inside breathing periodical structures in a two-variable model of a one-dimensional infinite excitable reaction-diffusion system.

A two-variable model of a one-dimensional (1D), infinite, excitable, reaction-diffusion system describing oscillons localized inside an expanding breathing periodical structure emitting traveling impulses is presented. The model is based on two coupled catalytic (enzymatic) reactions.

Stationary periodical structure emitting an infinite number of traveling impulses in a model of a one-dimensional infinite excitable reaction-diffusion system.

A two-variable model of a one-dimensional infinite excitable reaction-diffusion system describing an expanding stationary periodical structure emitting traveling impulses is presented. The model is based on two coupled catalytic (enzymatic) reactions. The chemical scheme consists of mono- and bimolecular reactions.

Stochastic transitions through unstable limit cycles in a model of bistable thermochemical system.

The master equation approach is used to study transitions through an unstable limit cycle surrounding a stable focus in two-variable systems with three stationary states. The model considered describes a bistable thermochemical system. Two cases are studied.

Master equation simulations of bistable and excitable dynamics in a model of a thermochemical system.

The effect of fluctuations on the dynamics of a model of a bistable thermochemical system is studied by means of the master equation. The system has three stationary states and exhibits two types of bistability: the coexistence of two stable focuses and the coexistence of a stable focus with a stable limit cycle separated by a saddle point. Stochastic effects are important when the system is close to the bifurcation, in which the stable limit cycle disappears through a homoclinic orbit.

Periodic spatiotemporal patterns in a two-dimensional two-variable reaction-diffusion model.

Periodic spatiotemporal two-dimensional (2D) asymptotic patterns in an excitable two-variable thermochemical (reaction-diffusion) system are shown. In a one-dimensional system the traveling impulse which reflects from impermeable boundaries is a stable asymptotic solution if the diffusion coefficient of the reactant is greater than the thermal diffusivity of the system. Periodic patterns of two symmetries are presented in the 2D system: the impulse of excitation propagating along the diagonal of a square spatial domain and a structure consisting of curved impulses which propagate in the direction perpendicular to one side of a rectangular domain.

On the variety of traveling fronts in one-variable multistable reaction-diffusion systems.

The still-open problem of the variety of asymptotic solutions to one-variable, one-dimensional infinite multistable reaction-diffusion systems is solved. We show that in such systems, besides monotonic traveling fronts, nonmonotonic traveling fronts can exist for appropriate initial conditions. The dependence of numbers of various types of traveling fronts on the number of stable stationary states also is given.

Breaking of translational symmetry of a traveling planar impulse in a two-dimensional two-variable reaction-diffusion model.

The stability of a planar impulse in rectangular spatial domains for a two-variable excitable reaction-diffusion system is numerically studied. The dependence of the stability on the size of the domain perpendicular to the direction of the propagation of the impulse is shown. The instability results in asymptotic stable curved impulses or an asymptotic spatiotemporal structure, which is generated similarly to the one-dimensional backfiring phenomenon.

Periodical survival or decay of traveling impulse in a model of a one-dimensional reaction-diffusion system.

A two-variable model of a one-dimensional (1D), open, excitable reaction-diffusion system describing space-time evolutions of traveling impulses is investigated. It is shown that depending on the size of the system, the traveling impulse can survive or decay. Continuous increase of the size of the system causes periodical repetitions of surviving and decay of the impulse.

Multipeak distributions of first passage times in bistable dynamics in a model of a thermochemical system.

A master equation is used to study transitions between the stable limit cycle and stable focus in the two-variable bistable system. The distribution function of the mean first passage time between these attractors and the relative dispersion of the mean first return time from the stable focus to itself as a function of the intensity of fluctuations are calculated and discussed. A coherence resonance is observed for the return time from the focus to itself.

Master equation simulations of a model of a thermochemical system.

Master equation approach is used to study the influence of fluctuations on the dynamics of a model thermochemical system. For appropriate values of parameters, the deterministic description of the system gives the subcritical or supercritical Hopf bifurcations. For small systems (containing 100 000 particles) close to the supercritical Hopf bifurcation, the stochastic trajectories obtained from numerical simulations do not allow to distinguish between damped oscillations around a stable focus and sustained oscillations around a small stable limit cycle.

Complex mixed-mode periodic and chaotic oscillations in a simple three-variable model of nonlinear system.

A detailed study of a generic model exhibiting new type of mixed-mode oscillations is presented. Period doubling and various period adding sequences of bifurcations are observed. New type of a family of 1D (one-dimensional) return maps is found.

Scalings of mixed-mode regimes in a simple polynomial three-variable model of nonlinear dynamical systems.

We describe scaling laws for a control parameter for various sequences of bifurcations of the LSn mixed-mode regimes consisting of single large amplitude maximum followed by n small amplitude peaks. These regimes are obtained in a normalized version of a simple three-variable polynomial model that contains only one nonlinear cubic term. The period adding bifurcations for LSn patterns scales as 1/n at low n and as 1/n2 at sufficiently large values of n.